Study of Several Lattice Systems with Long-Range Forces

Abstract

A number of one- and two-dimensional Ising lattice systems with long-range ferromagnetic interactions are studied. The theory introduces as basic variables stochastic fields acting at each site, but goes beyond Weiss mean-field theory (or the Bragg-Williams approximation) in giving a complete account of the statistics of these fields. A transition is manifest in these systems by a shift in the values of the stochastic fields which are important for the calculation of the partition function. Particular attention is devoted to the critical region where the range of significant stochastic fields broadens. The equation of state for the lattice gas corresponding to this model is of the van der Waals type. Comparison is frequently made between these results and the properties of an analogous one-dimensional continuum system studied by Kac, Uhlenbeck, and Hemmer.